A salesman's commission is 5% on all sales up to ₹ 10000 and 4% of all sales exceeding this amount. He remits ₹ 31100 to the parent company after deducting his commission. His sales were worth

### Concept & Equation The total sales amount is divided into two tiers for commission calculation. The remittance to the parent company is strictly the total sales minus the total commission earned. $$\text{Remittance} = \text{Total Sales} - \text{Total Commission}$$ ### Step-by-Step Solution * Let the total sales be $S$. * The commission on the first ₹ 10000 is 5%: $$C_1 = 0.05 \times 10000 = 500$$ * The commission on sales exceeding ₹ 10000 is 4%: $$C_2 = 0.04 \times (S - 10000)$$ * Set up the remittance equation based on total sales minus total commission: $$31100 = S - [500 + 0.04(S - 10000)]$$ * Distribute and simplify the equation: $$31100 = S - 500 - 0.04S + 400$$ $$31100 = 0.96S - 100$$ * Solve for $S$: $$31200 = 0.96S$$ $$S = \frac{31200}{0.96} = \frac{3120000}{96} = 32500$$ ### Exam Strategy & Shortcut Instead of heavy algebra, logically shift the commission rate. If the salesman earned a flat 4% on *all* sales, his commission on the first ₹ 10000 would drop by 1% (from 5% to 4%). 1% of 10000 = ₹ 100. If his commission drops by ₹ 100, his remittance to the company must increase by ₹ 100. New theoretical remittance = $31100 + 100 = 31200$. Since the company now gets 96% of total sales: $$0.96S = 31200 \implies S = 32500$$. This mental shift avoids expanding brackets entirely. ### Common Pitfall A major error is formulating the commission on the remaining amount as just $0.04S$ instead of $0.04(S - 10000)$. This double-counts the first ₹ 10000 and heavily skews the final algebraic result. ### Final Answer **Therefore, the correct answer is ₹ 32500.**